A modified direct method is developed to find finite symmetry groups of nonlinear mathematical physics systems. Applying the modified direct method to the well-known （2＋1）-dimensional asymmetric Nizhnik-Novikov-Ves...A modified direct method is developed to find finite symmetry groups of nonlinear mathematical physics systems. Applying the modified direct method to the well-known （2＋1）-dimensional asymmetric Nizhnik-Novikov-Vesselov equation and Nizhnik Novikov-Vesselov equation, both the Lie point symmetry groups and the non-Lie symmetry groups are obtained. The Lie symmetry groups obtained via traditional Lie approaches are only speciai cases. Furthermore, the expressions of the exact finite transformations of the Lie groups are much simpler than those obtained via the standard approaches.展开更多
We give the generalized definitions of variable separable solutions to nonlinear evolution equations, and characterize the relation between the functional separable solution and the derivative-dependent functional sep...We give the generalized definitions of variable separable solutions to nonlinear evolution equations, and characterize the relation between the functional separable solution and the derivative-dependent functional separable solution. The new definitions can unify various kinds of variable separable solutions appearing in references. As application, we classify the generalized nonlinear diffusion equations that admit special functional separable solutions and obtain some exact solutions to the resulting equations.展开更多
The group classification is carried out on the nonlinear wave equation utt = f（x,u, ux）uzz ＋ g（x,u,uz） by using the preliminary group classification approach. The generators of equivalence group are determined an...The group classification is carried out on the nonlinear wave equation utt = f（x,u, ux）uzz ＋ g（x,u,uz） by using the preliminary group classification approach. The generators of equivalence group are determined and the corresponding reduced forms are obtained. The result of the work is shown in table form.展开更多
Using the sign-invariant theory, we study the nonlinear reaction-diffusion systems. We also obtain some new explicit solutions to the nonlinear resulting systems.
Some extended solution mapping relations of the nonlinear coupled scalar field and the well-known φ^4 model are presented. Simultaneously, inspired by the new solutions of the famous φ^4 model recently proposed by J...Some extended solution mapping relations of the nonlinear coupled scalar field and the well-known φ^4 model are presented. Simultaneously, inspired by the new solutions of the famous φ^4 model recently proposed by Jia, Huang and Lou, five kinds of new localized excitations of the nonlinear coupled scaiar field （NCSF） system are obtained.展开更多
It is difficult to obtain exact solutions of the nonlinear partial differential equations(PDEs)due to their complexity and nonlinearity,especially for non-integrable systems.In this paper,some reasonable approximation...It is difficult to obtain exact solutions of the nonlinear partial differential equations(PDEs)due to their complexity and nonlinearity,especially for non-integrable systems.In this paper,some reasonable approximations of real physics are considered,and the invariant expansion is proposed to solve real nonlinear systems.A simple invariant expansion with quite a universal pseudopotential is used for some nonlinear PDEs such as the Korteweg-de Vries(KdV)equation with a fifth-order dispersion term,the perturbed fourth-order KdV equation,the KdV-Burgers equation,and a Boussinesq-type equation.展开更多
Transformer models have emerged as pivotal tools within the realm of drug discovery,distinguished by their unique architectural features and exceptional performance in managing intricate data landscapes.Leveraging the...Transformer models have emerged as pivotal tools within the realm of drug discovery,distinguished by their unique architectural features and exceptional performance in managing intricate data landscapes.Leveraging the innate capabilities of transformer architectures to comprehend intricate hierarchical dependencies inherent in sequential data,these models showcase remarkable efficacy across various tasks,including new drug design and drug target identification.The adaptability of pre-trained trans-former-based models renders them indispensable assets for driving data-centric advancements in drug discovery,chemistry,and biology,furnishing a robust framework that expedites innovation and dis-covery within these domains.Beyond their technical prowess,the success of transformer-based models in drug discovery,chemistry,and biology extends to their interdisciplinary potential,seamlessly combining biological,physical,chemical,and pharmacological insights to bridge gaps across diverse disciplines.This integrative approach not only enhances the depth and breadth of research endeavors but also fosters synergistic collaborations and exchange of ideas among disparate fields.In our review,we elucidate the myriad applications of transformers in drug discovery,as well as chemistry and biology,spanning from protein design and protein engineering,to molecular dynamics(MD),drug target iden-tification,transformer-enabled drug virtual screening(VS),drug lead optimization,drug addiction,small data set challenges,chemical and biological image analysis,chemical language understanding,and single cell data.Finally,we conclude the survey by deliberating on promising trends in transformer models within the context of drug discovery and other sciences.展开更多
RNAs have important biological functions and the functions of RNAs are generally coupled to their structures, especiallytheir secondary structures. In this work, we have made a comprehensive evaluation of the performa...RNAs have important biological functions and the functions of RNAs are generally coupled to their structures, especiallytheir secondary structures. In this work, we have made a comprehensive evaluation of the performances of existingtop RNA secondary structure prediction methods, including five deep-learning (DL) based methods and five minimum freeenergy (MFE) based methods. First, we made a brief overview of these RNA secondary structure prediction methods.Afterwards, we built two rigorous test datasets consisting of RNAs with non-redundant sequences and comprehensivelyexamined the performances of the RNA secondary structure prediction methods through classifying the RNAs into differentlength ranges and different types. Our examination shows that the DL-based methods generally perform better thanthe MFE-based methods for RNAs with long lengths and complex structures, while the MFE-based methods can achievegood performance for small RNAs and some specialized MFE-based methods can achieve good prediction accuracy forpseudoknots. Finally, we provided some insights and perspectives in modeling RNA secondary structures.展开更多
Duality analysis of time series and complex networks has been a frontier topic during the last several decades.According to some recent approaches in this direction,the intrinsic dynamics of typical nonlinear systems ...Duality analysis of time series and complex networks has been a frontier topic during the last several decades.According to some recent approaches in this direction,the intrinsic dynamics of typical nonlinear systems can be better characterized by considering the related nonlinear time series from the perspective of networks science.In this paper,the associated network family of the unified piecewise-linear(PWL)chaotic family,which can bridge the gap of the PWL chaotic Lorenz system and the PWL chaotic Chen system,was firstly constructed and analyzed.We constructed the associated network family via the original and the modified frequency-degree mapping strategy,as well as the classical visibility graph and horizontal visibility graph strategy,after removing the transient states.Typical related network characteristics,including the network fractal dimension,of the associated network family,are computed with changes of single key parameter a.These characteristic vectors of the network are also compared with the largest Lyapunov exponent(LLE)vector of the related original dynamical system.It can be found that,some network characteristics are highly correlated with LLE vector of the original nonlinear system,i.e.,there is an internal consistency between the largest Lyapunov exponents,some typical associated network characteristics,and the related network fractal dimension index.Numerical results show that the modified frequency-degree mapping strategy can demonstrate highest correlation,which means it can behave better to capture the intrinsic characteristics of the unified PWL chaotic family.展开更多
The concept of approximate generalized conditional symmetry （A GCS） as a generalization to both approximate Lie point symmetry and generalized conditional symmetry is introduced, and it is applied to study the pertu...The concept of approximate generalized conditional symmetry （A GCS） as a generalization to both approximate Lie point symmetry and generalized conditional symmetry is introduced, and it is applied to study the perturbed nonlinear diffusion-convection equations. Complete classification of those perturbed equations which admit cerrain types of AGCSs is derived. Some approximate solutions to the resulting equations can be obtained via the AGCS and the corresponding unperturbed equations.展开更多
The equivalence of three (2 + 1)-dimensional soliton equations is proved, and the quite generalsolutionswitha some arbitrary functions of x, t and y respectively are obtained. By selecting the arbitrary functions, man...The equivalence of three (2 + 1)-dimensional soliton equations is proved, and the quite generalsolutionswitha some arbitrary functions of x, t and y respectively are obtained. By selecting the arbitrary functions, many specialtypes of the localized excitations like the solitoff solitons, multi-dromion solutions, lump, and multi-ring soliton solutionsare obtained.展开更多
The weak Darboux transformation of the （2＋1） dimensional Euler equation is used to find its exact solutions. Starting from a constant velocity field solution, a set of quite general periodic wave solutions such as ...The weak Darboux transformation of the （2＋1） dimensional Euler equation is used to find its exact solutions. Starting from a constant velocity field solution, a set of quite general periodic wave solutions such as the Rossby waves can be simply obtained from the weak Darboux transformation with zero spectral parameters. The constant vorticity seed solution is utilized to generate Bessel waves.展开更多
This paper is devoted to the study of functional variable separation for extended nonlinear elliptic equations. By applying the functional variable separation approach to extended nonlinear elliptic equations via the ...This paper is devoted to the study of functional variable separation for extended nonlinear elliptic equations. By applying the functional variable separation approach to extended nonlinear elliptic equations via the generalized conditional symmetry, we obtain complete classification of those equations which admit functional separable solutions （FSSs） and construct some exact FSSs to the resulting equations.展开更多
This paper is devoted to studying symmetry reduction of Cauchy problems for the fourth-order quasi-linear parabolic equations that admit certain generalized conditional symmetries （GCSs）. Complete group classificati...This paper is devoted to studying symmetry reduction of Cauchy problems for the fourth-order quasi-linear parabolic equations that admit certain generalized conditional symmetries （GCSs）. Complete group classification results are presented, and some examples are given to show the main reduction procedure.展开更多
In terms of our new exact definition of partial Lagrangian and approximate Euler-Lagrange-type equation, we investigate the nonlinear wave equation with damping via approximate Noether-type symmetry operators associat...In terms of our new exact definition of partial Lagrangian and approximate Euler-Lagrange-type equation, we investigate the nonlinear wave equation with damping via approximate Noether-type symmetry operators associated with partial Lagrangians and construct its approximate conservation laws in general form.展开更多
This paper studies the perturbed nonlinear diffusion-convection equation with source term via the approximate generalized conditional symmetry （A GCS）. Complete classification of those perturbed equations which admi...This paper studies the perturbed nonlinear diffusion-convection equation with source term via the approximate generalized conditional symmetry （A GCS）. Complete classification of those perturbed equations which admit certain types of AGCSs is derived. Some approximate invariant solutions to the resulting equations can also be obtained.展开更多
Symmetry reduction of a class of third-order evolution equations that admit certain generalized conditionalsymmetries (GCSs) is implemented.The reducibility of the initial-value problem for an evolution equation to a ...Symmetry reduction of a class of third-order evolution equations that admit certain generalized conditionalsymmetries (GCSs) is implemented.The reducibility of the initial-value problem for an evolution equation to a Cauchyproblem for a system of ordinary differential equations (ODEs) is characterized via the GCS and its Lie symmetry.Complete classification theorems are obtained and some examples are taken to show the main reduction procedure.展开更多
We exploit higher-order conditional symmetry to reduce initial-value problems for evolution equations toCauchy problems for systems of ordinary differential equations (ODEs).We classify a class of fourth-order evoluti...We exploit higher-order conditional symmetry to reduce initial-value problems for evolution equations toCauchy problems for systems of ordinary differential equations (ODEs).We classify a class of fourth-order evolutionequations which admit certain higher-order generalized conditional symmetries (GCSs) and give some examples to showthe main reduction procedure.These reductions cannot be derived within the framework of the standard Lie approach,which hints that the technique presented here is something essential for the dimensional reduction of evolu tion equations.展开更多
Approximate generalized conditional symmetry is developed to study the approximate symmetry reduction for initial-value problems of the extended KdV-Burgers equations with perturbation.These equations can be reduced t...Approximate generalized conditional symmetry is developed to study the approximate symmetry reduction for initial-value problems of the extended KdV-Burgers equations with perturbation.These equations can be reduced to initial-value problems for some systems of first-order perturbed ordinary differential equations in terms of a new approach.Complete classification theorems are obtained and an example is taken to show the main reduction procedure.展开更多
The structural and electronic properties of monovacancy,divacancy defects within crystalline silicon have been investigated systematically using a new tight-binding model with a 216-atom supercell.The formation energi...The structural and electronic properties of monovacancy,divacancy defects within crystalline silicon have been investigated systematically using a new tight-binding model with a 216-atom supercell.The formation energies and energy levels of all the defect configurations are carefully calculated.The results show that atoms nearer to the defects naturally contribute to gap states more,and are comparable with the experimental values.展开更多
基金The project supported by the National 0utstanding Youth Foundation of China under Grant No. 19925522 and the National Natural Science Foundation of China under Grant Nos. 90203001, 10475055. The authors are in debt to thank helpful discussions with Drs. X.Y. Tang, C.L. Chen, Y. Chen, H.C. Hu, X.M. Qian, B. Tong, and W.R. Cai.
文摘A modified direct method is developed to find finite symmetry groups of nonlinear mathematical physics systems. Applying the modified direct method to the well-known （2＋1）-dimensional asymmetric Nizhnik-Novikov-Vesselov equation and Nizhnik Novikov-Vesselov equation, both the Lie point symmetry groups and the non-Lie symmetry groups are obtained. The Lie symmetry groups obtained via traditional Lie approaches are only speciai cases. Furthermore, the expressions of the exact finite transformations of the Lie groups are much simpler than those obtained via the standard approaches.
基金The project supported by National Natural Science Foundation of China under Grant Nos.10447007 and 10671156the Natural Science Foundation of Shaanxi Province of China under Grant No.2005A13
文摘We give the generalized definitions of variable separable solutions to nonlinear evolution equations, and characterize the relation between the functional separable solution and the derivative-dependent functional separable solution. The new definitions can unify various kinds of variable separable solutions appearing in references. As application, we classify the generalized nonlinear diffusion equations that admit special functional separable solutions and obtain some exact solutions to the resulting equations.
基金Supported by NSF-China Grant 10671156NSF of Shaanxi Province of China (SJ08A05) NWU Graduate Innovation and Creativity Funds under Grant No.09YZZ56
文摘The group classification is carried out on the nonlinear wave equation utt = f（x,u, ux）uzz ＋ g（x,u,uz） by using the preliminary group classification approach. The generators of equivalence group are determined and the corresponding reduced forms are obtained. The result of the work is shown in table form.
基金National Natural Science Foundation of China under Grant Nos.10447007 and 10671156the Natural Science Foundation of Shaanxi Province of China under Grant No.2005A13
文摘Using the sign-invariant theory, we study the nonlinear reaction-diffusion systems. We also obtain some new explicit solutions to the nonlinear resulting systems.
基金National Natural Science Foundation of China under Grant Nos.10475055 and 90503006the Scientific Research Fund of the Education Department of Zhejiang Province under Grant No.20040969
文摘Some extended solution mapping relations of the nonlinear coupled scalar field and the well-known φ^4 model are presented. Simultaneously, inspired by the new solutions of the famous φ^4 model recently proposed by Jia, Huang and Lou, five kinds of new localized excitations of the nonlinear coupled scaiar field （NCSF） system are obtained.
基金Project supported by the National Natural Science Foundation of China(Grant No.11175092)Scientific Research Fund of Zhejiang Provincial Education Department(Grant No.Y201017148)K.C.Wong Magna Fund in Ningbo University
文摘It is difficult to obtain exact solutions of the nonlinear partial differential equations(PDEs)due to their complexity and nonlinearity,especially for non-integrable systems.In this paper,some reasonable approximations of real physics are considered,and the invariant expansion is proposed to solve real nonlinear systems.A simple invariant expansion with quite a universal pseudopotential is used for some nonlinear PDEs such as the Korteweg-de Vries(KdV)equation with a fifth-order dispersion term,the perturbed fourth-order KdV equation,the KdV-Burgers equation,and a Boussinesq-type equation.
基金supported in part by National Institute of Health(NIH),USA(Grant Nos.:R01GM126189,R01AI164266,and R35GM148196)the National Science Foundation,USA(Grant Nos.DMS2052983,DMS-1761320,and IIS-1900473)+3 种基金National Aero-nautics and Space Administration(NASA),USA(Grant No.:80NSSC21M0023)Michigan State University(MSU)Foundation,USA,Bristol-Myers Squibb(Grant No.:65109)USA,and Pfizer,USAsupported by the National Natural Science Foundation of China(Grant Nos.:11971367,12271416,and 11972266).
文摘Transformer models have emerged as pivotal tools within the realm of drug discovery,distinguished by their unique architectural features and exceptional performance in managing intricate data landscapes.Leveraging the innate capabilities of transformer architectures to comprehend intricate hierarchical dependencies inherent in sequential data,these models showcase remarkable efficacy across various tasks,including new drug design and drug target identification.The adaptability of pre-trained trans-former-based models renders them indispensable assets for driving data-centric advancements in drug discovery,chemistry,and biology,furnishing a robust framework that expedites innovation and dis-covery within these domains.Beyond their technical prowess,the success of transformer-based models in drug discovery,chemistry,and biology extends to their interdisciplinary potential,seamlessly combining biological,physical,chemical,and pharmacological insights to bridge gaps across diverse disciplines.This integrative approach not only enhances the depth and breadth of research endeavors but also fosters synergistic collaborations and exchange of ideas among disparate fields.In our review,we elucidate the myriad applications of transformers in drug discovery,as well as chemistry and biology,spanning from protein design and protein engineering,to molecular dynamics(MD),drug target iden-tification,transformer-enabled drug virtual screening(VS),drug lead optimization,drug addiction,small data set challenges,chemical and biological image analysis,chemical language understanding,and single cell data.Finally,we conclude the survey by deliberating on promising trends in transformer models within the context of drug discovery and other sciences.
基金supported by grants from the National Science Foundation of China(Grant Nos.12375038 and 12075171 to ZJT,and 12205223 to YLT).
文摘RNAs have important biological functions and the functions of RNAs are generally coupled to their structures, especiallytheir secondary structures. In this work, we have made a comprehensive evaluation of the performances of existingtop RNA secondary structure prediction methods, including five deep-learning (DL) based methods and five minimum freeenergy (MFE) based methods. First, we made a brief overview of these RNA secondary structure prediction methods.Afterwards, we built two rigorous test datasets consisting of RNAs with non-redundant sequences and comprehensivelyexamined the performances of the RNA secondary structure prediction methods through classifying the RNAs into differentlength ranges and different types. Our examination shows that the DL-based methods generally perform better thanthe MFE-based methods for RNAs with long lengths and complex structures, while the MFE-based methods can achievegood performance for small RNAs and some specialized MFE-based methods can achieve good prediction accuracy forpseudoknots. Finally, we provided some insights and perspectives in modeling RNA secondary structures.
文摘Duality analysis of time series and complex networks has been a frontier topic during the last several decades.According to some recent approaches in this direction,the intrinsic dynamics of typical nonlinear systems can be better characterized by considering the related nonlinear time series from the perspective of networks science.In this paper,the associated network family of the unified piecewise-linear(PWL)chaotic family,which can bridge the gap of the PWL chaotic Lorenz system and the PWL chaotic Chen system,was firstly constructed and analyzed.We constructed the associated network family via the original and the modified frequency-degree mapping strategy,as well as the classical visibility graph and horizontal visibility graph strategy,after removing the transient states.Typical related network characteristics,including the network fractal dimension,of the associated network family,are computed with changes of single key parameter a.These characteristic vectors of the network are also compared with the largest Lyapunov exponent(LLE)vector of the related original dynamical system.It can be found that,some network characteristics are highly correlated with LLE vector of the original nonlinear system,i.e.,there is an internal consistency between the largest Lyapunov exponents,some typical associated network characteristics,and the related network fractal dimension index.Numerical results show that the modified frequency-degree mapping strategy can demonstrate highest correlation,which means it can behave better to capture the intrinsic characteristics of the unified PWL chaotic family.
基金Supported by the National Natural Science Foundation of China under Grant Nos 10371098 and 10447007, the Natural Science Foundation of Shaanxi Province （No 2005A13）, and the Special Research Project of Educational Department of Shaanxi Province （No 03JK060）.
文摘The concept of approximate generalized conditional symmetry （A GCS） as a generalization to both approximate Lie point symmetry and generalized conditional symmetry is introduced, and it is applied to study the perturbed nonlinear diffusion-convection equations. Complete classification of those perturbed equations which admit cerrain types of AGCSs is derived. Some approximate solutions to the resulting equations can be obtained via the AGCS and the corresponding unperturbed equations.
文摘The equivalence of three (2 + 1)-dimensional soliton equations is proved, and the quite generalsolutionswitha some arbitrary functions of x, t and y respectively are obtained. By selecting the arbitrary functions, many specialtypes of the localized excitations like the solitoff solitons, multi-dromion solutions, lump, and multi-ring soliton solutionsare obtained.
基金Supported by the National Natural Science Foundation of China under Grant Nos 90203001, 10475055 and 90503006.
文摘The weak Darboux transformation of the （2＋1） dimensional Euler equation is used to find its exact solutions. Starting from a constant velocity field solution, a set of quite general periodic wave solutions such as the Rossby waves can be simply obtained from the weak Darboux transformation with zero spectral parameters. The constant vorticity seed solution is utilized to generate Bessel waves.
基金The project supported by National Natural Science Foundation of China under Grant No. 10447007 and the Natural Science Foundation of Shaanxi Province of China under Grant No. 2005A13
文摘This paper is devoted to the study of functional variable separation for extended nonlinear elliptic equations. By applying the functional variable separation approach to extended nonlinear elliptic equations via the generalized conditional symmetry, we obtain complete classification of those equations which admit functional separable solutions （FSSs） and construct some exact FSSs to the resulting equations.
基金Supported by the National Natural Science Foundation of China under Grant No.10671156the Natural Science Foundation of Shaanxi Province of China under Grant No.SJ08A05
文摘This paper is devoted to studying symmetry reduction of Cauchy problems for the fourth-order quasi-linear parabolic equations that admit certain generalized conditional symmetries （GCSs）. Complete group classification results are presented, and some examples are given to show the main reduction procedure.
基金Supported by the National Natural Science Foundation of China under Grant No.10671156the Natural Science Foundation of Shaanxi Province of China under Grant No.SJ08A05
文摘In terms of our new exact definition of partial Lagrangian and approximate Euler-Lagrange-type equation, we investigate the nonlinear wave equation with damping via approximate Noether-type symmetry operators associated with partial Lagrangians and construct its approximate conservation laws in general form.
基金The project supported by National Natural Science Foundation of China under Grant Nos.10371098 and 10447007the Natural Science Foundation of Shanxi Province of China under Grant No.2005A13
文摘This paper studies the perturbed nonlinear diffusion-convection equation with source term via the approximate generalized conditional symmetry （A GCS）. Complete classification of those perturbed equations which admit certain types of AGCSs is derived. Some approximate invariant solutions to the resulting equations can also be obtained.
基金Supported by the National Natural Science Foundation of China under Grant Nos.10447007 and 10671156the Natural Science Foundation of Shaanxi Province of China under Grant No.2005A13
文摘Symmetry reduction of a class of third-order evolution equations that admit certain generalized conditionalsymmetries (GCSs) is implemented.The reducibility of the initial-value problem for an evolution equation to a Cauchyproblem for a system of ordinary differential equations (ODEs) is characterized via the GCS and its Lie symmetry.Complete classification theorems are obtained and some examples are taken to show the main reduction procedure.
基金National Natural Science Foundation of China under Grant Nos.10447007 and 10671156the Natural Science Foundation of Shaanxi Province of China under Grant No.2005A13
文摘We exploit higher-order conditional symmetry to reduce initial-value problems for evolution equations toCauchy problems for systems of ordinary differential equations (ODEs).We classify a class of fourth-order evolutionequations which admit certain higher-order generalized conditional symmetries (GCSs) and give some examples to showthe main reduction procedure.These reductions cannot be derived within the framework of the standard Lie approach,which hints that the technique presented here is something essential for the dimensional reduction of evolu tion equations.
基金Supported by the National Natural Science Foundation of China under Grant No 10671156the Natural Science Foundation of Shaanxi Province of China under Grant No SJ08A05the NWU Graduate Innovation and Creativity Funds(09YZZ56)。
文摘Approximate generalized conditional symmetry is developed to study the approximate symmetry reduction for initial-value problems of the extended KdV-Burgers equations with perturbation.These equations can be reduced to initial-value problems for some systems of first-order perturbed ordinary differential equations in terms of a new approach.Complete classification theorems are obtained and an example is taken to show the main reduction procedure.
基金Supported by the National Natural Science Foundation of China under Grant No.69876035,the Fund of Chinese Academy of Sciences and the Fund of University of Science and Technology of China.
文摘The structural and electronic properties of monovacancy,divacancy defects within crystalline silicon have been investigated systematically using a new tight-binding model with a 216-atom supercell.The formation energies and energy levels of all the defect configurations are carefully calculated.The results show that atoms nearer to the defects naturally contribute to gap states more,and are comparable with the experimental values.
